Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

Stability of the vortex defect in Landau - de Gennes theory of nematic liquid crystals

Radu Ignat

, Luc Nguyen

Valeriy Slastikov

Arghir Zarnescu

aLaboratoire de Math´ematiques, Universit´e Paris-Sud 11, bˆat. 425, 91405 Orsay, France

bMathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA

cSchool of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

dUniversity of Sussex, Department of Mathematics, Pevensey 2, Falmer, BN1 9QH, United Kingdom

Received *****; accepted after revision +++++

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Abstract

We analyze the radially symmetric solution corresponding to the vortex defect (the so calledmelting hedgehog) in Landau - de Gennes theory for nematic liquid crystals. We prove existence, uniqueness and stability results of the melting hedgehog.To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

R´esum´e

Stabilit´e du d´efaut vortex dans la th´eorie Landau - de Gennes pour les cristaux liquides. Nous

´

etudions la solution `a sym´etrie radiale associ´ee au d´efaut de type vortex dans la th´eorie Landau - de Gennes pour les cristaux liquides. Nous montrons des r´esultats d’existence, unicit´e et stabilit´e de cette solution.Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Version fran¸caise abr´eg´ee

Dans cette note, nous ´etudions la fonctionnelle de Landau - de Gennes (LdG) pour les cristaux liquides dont le param`etre d’ordre Q(x) est d´efini en tout point x ∈ R<sup>3</sup> et `a valeurs dans l’espace S0 des Q- tenseurs de dimension 5, i.e., les matrices sym´etriques 3×3 `a trace nulle. L’´energie LdG est constitu´ee de deux termes : l’´energie ´elastique et l’´energie potentielle dont la densit´e est donn´ee par un polynˆomef_B(Q) de degr´e 4 en Q(voir (1.1)) qui atteint sa valeur minimale sur une sous-vari´et´e de S0 de dimension 2.

Notre but consiste `a analyser le point critique de cette fonctionnelle qui a une sym´etrie radiale et satisfait des conditions limites correspondant au d´efaut vortex. Cette solution s’appelle “melting hedgehog” et s’´ecrit sous la formeH(x) =u(|x|) ¯H(x) (voir (1.3)) o`u le profil radial usatisfait une EDO semilin´eaire d’ordre deux (voir (1.4)) avec des conditions limites `a l’origine 0 et `a l’infini. En effet, ¯H(x) repr´esente le Q-tenseur associ´e au champ de vortex singulier x/|x| dans la th´eorie Ginzburg-Landau en 3D et la fonction scalaireu(|x|) correspond au profil (scalaire) r´egulier du vortex qui s’annule `a l’origine.

D’abord, nous montrons l’existence et l’unicit´e du profil radial positif u. La preuve de l’existence du profil u repose sur une approche variationnelle, tandis que l’unicit´e est bas´ee sur un principe de

Email addresses:Radu [email protected](Radu Ignat),[email protected](Luc Nguyen), [email protected](Valeriy Slastikov),[email protected](Arghir Zarnescu).

comparaison pour les sur/sous-solutions de l’EDO (1.4) et une analyse du comportement asymptotique de la solution en 0 et `a l’∞.

Ensuite, nous pr´esentons le r´esultat principal de cette note sur la stabilit´e / instabilit´e de la solution H selon le param`etrea<sup>2</sup> du syst`eme. Premi`erement, nous montrons l’instabilit´e deH dans le r´egime o`u a²est grand (correspondant physiquement `a une basse temp´erature r´eduite). Ceci repose sur la construc- tion d’une perturbation autour de la solution H qui rend n´egative la variation seconde de l’´energie.

Deuxi`emement, nous prouvons la stabilit´e deH dans le r´egime o`u le param`etre a<sup>2</sup> est petit. En effet, nous montrons la positivit´e de la variation secondeQ(V) pour toute perturbation V autour deH : nous utilisons d’abord une d´ecomposition de V en une base bien choisie dans S0 et ensuite, la m´ethode de la s´eparation de variable pour r´eduire d’abord le probl`eme au contexte `a sym´etrie axiale et ensuite, au contexte radial. La conclusion suit par un argument de type Hardy qui rend compte de la nonlin´earit´e f<sub>B</sub>(Q) du syst`eme.

1. Introduction

We consider the following Landau-de Gennes energy functional:

F(Q) = Z

R³

h1

2|∇Q|²+f_B(Q)i

dx, Q∈H¹(R³,S0), where

S0

def={Q∈R^3×3, Q=Q^t,tr(Q) = 0}

denotes the set of Q-tensors (i.e., traceless symmetric matrices inR^3×3, see [1] for their physical inter- pretation). The bulk energy densityfB accounts for the bulk effects and has the following form

fB(Q) =−a²

2 |Q|²−b²

3tr(Q³) +c²

4|Q|⁴, (1.1)

wherea², c²>0 andb²≥0 are constants and|Q|^2def= tr(Q²). A critical point of the functionalF satisfies the Euler-Lagrange equation

∆Q=−a²Q−b²[Q²−1

3|Q|²Id] +c²|Q|²Q, (1.2) where the term ¹₃b²|Q|²Id is a Lagrange multiplier that accounts for the tracelessness constraint. It is well known that solutions of (1.2) are smooth (see for instance [8]).

Remark 1 We should point out now that although the equation (1.2)seems to depend on three parameters a², b² and c², there is only one independent parameter in the problem which can be chosen to be a² for fixedb² andc² (after a suitable rescaling Q7→λQ(x/µ)for two parametersλ, µ >0).

We are interested in studying theradially-symmetricsolution of (1.2). For that, a measurableS0-valued mapQ:R³→S0 is called radially-symmetric if

Q(Rx) =R Q(x)R^tfor any rotationR∈SO(3) and a.e.x∈R³.

In fact, such a mapQ(x) has only one degree of freedom: there exists a measurable radial scalar function u: (0,+∞)→Rsuch thatQ(x) =u(|x|) H(x) for a.e.x∈R³, where H is the so calledhedgehog:

H(x) = x

|x|⊗ x

|x|−1 3Id

and the radial scalar profileuofQis given byu(|x|) =³₂tr(Q(x)H(x)) for a.e.x∈R³.

We will focus on the stability and properties of the profile of the following radially symmetric solution of (1.2), calledmelting hedgehog:

H(x) =u(|x|)H(x) (1.3)

where the radial scalar profileuis the uniquepositivesolution of the following semilinear ODE:

u⁰⁰(r) +2

ru⁰(r)− 6

r²u(r) =F(u(r)) for r >0 (1.4) subject to boundary conditionsu(0) = 0 and lim_r→∞u(r) =s+ where

F(u) =−a²u−b²

3 u²+2c²

3 u³ (1.5)

ands+

def= ^b2+

√b⁴+24a²c²

4c² is the positive zero ofF.

Our main result concerns the local stability of the melting hedgehog (see [6]). We highlight that H is a critical point of (1.2), but has infinite energyF, i.e., F(H) =∞. Therefore, the stability issue is carried out by analyzing the following second variation of the modified functionalF at the point H in the directionV ∈C_c^∞(R³;S0), denoted by Q(V):

Q(V) = 1 2

d² dt²

Z

R³

h1

2|∇(H+t V)|²+fB(H+t V)−1

2|∇H|²−fB(H)i dx

= Z

R³

h1

2|∇V|²+ −a²

2 +c²u² 3

|V|²−b²utr( ¯H V²) +c²u²tr²( ¯H V)i

dx. (1.6)

Theorem 1.1 There existsa²₀>0 such that for alla²< a²₀the melting hedgehogH defined at (1.3)is a locally stable critical point of (1.2), i.e.,Q(V)≥0 for all V ∈C_c^∞(R³;S0). MoreoverQ(V) = 0if and only ifV ∈span{∂_xiH}³_i=1, i.e., the kernel of the second variation is generated by translations ofH(x).

There exists a²₁>0 so that for any a²> a²₁ there exists V_∗ ∈C_c^∞(R³;S0)such that Q(V_∗)<0. Any suchV_∗ cannot be purely uniaxial (i.e.,V_∗(x)has three different eigenvalues for some point x∈R³).

2. Comparison with the Ginzburg-Landau model

Let us compare Landau- de Gennes (LdG) model with 2D and 3D Ginzburg-Landau (GL) systems [2,10,9]. Both LdG and GL energies consist of two terms: a Dirichlet energy and a well potential energy.

TheN-dimensional GL energy is defined forN-dimensional vector fields (here,N = 2,3) and the minima of the well potential is a co-dimension one manifold, e.g., the unit sphere S^N−1 ⊂ R^N. However, our 3D LdG energy is defined for maps with values into the five-dimensional space S0 ofQ-tensors and the global minimizingQ-tensors of the bulk potentialfB, as defined in (1.1), form a 2D manifoldM ⊂S0

(the limit target manifold), homeomorphic to the projective planeRP²⊂R³: M={Q∈S0 : Q=s₊

n⊗n−1 3Id

, n∈S²}. (2.1)

Thus, one might expect similar stability behavior between the melting hedgehog for the LdG and the vortex defect for the 3D GL (as maps defined from R³ into a 2D manifold), see [4]; nevertheless, there is still a significant difference to 3D GL since variations in LdG are allowed in 5D and the target limit manifolds (RP² for LdG and S² for 3D GL) have different fundamental groups. We highlight that the situation would be rather different ifb²= 0 in the expression offB of the LdG model: in that case, the global minimizingQ-tensors form a 4D sphere {Q∈S0 : tr(Q²) =a²/c²} hence we expect instability of the melting hedgehog as a map fromR³ with values into a 4Dmanifold. It was proved in [3] that this behavior holds true ifb²is small (corresponding toa² large after the rescaling in Remark 1). Our major

contribution is to develop a systematic approach (going beyond the Ginzburg-Landau type of methods) that is capable of dealing not only with the regime a² → ∞ but also with the much more challenging case whena²is small.

3. Existence and uniqueness of the radial scalar profile

We present now an existence and uniqueness result of the positive solution for a special type of semilinear ODE that generalizes (1.4). For that, letF :R+→Rbe aC¹function satisfying the following conditions:

s+ t F(t)

Figure 1. A schematic graph ofF onR+.







F(0) =F(s+) = 0, F⁰(s+)>0, F(t)<0 if t∈(0, s+),

F(t)>0 if t > s+,

(3.1) for somes+>0.

We will focus on the following semilinear ODE:

u⁰⁰(r) +p

ru⁰(r)− q

r²u(r) =F(u(r)) on (0, R) (3.2) where we assume

p, q∈Randq >0.

We will present the result for the case of finite domains (i.e., (0, R) with R∈(0,+∞)) and the infinite domain (0,+∞) (i.e.R= +∞) under the limit conditions

u(0) = 0, u(R) =s+, (3.3)

with the standard conventionu(∞) = limr→∞u(r) =s₊ ifR= +∞.

Theorem 3.1 Under the assumption (3.1), there exists a unique non-negative solution u of (3.2)with boundary conditions (3.3). Moreover, this solution is strictly increasing.

The existence result is proved by constructing positive energy minimizing solutions on finite intervals (0, R) satisfying (3.3). In the case ofR= +∞, we prove that those minimizers converge to a non-negative solution of (3.2) on the entire R+; the limit conditions (3.3) are satisfied due to the local minimizing behavior of the constructed solution that ensures non-flatness at +∞. The trickiest part consists in proving the uniqueness. In fact, the argument is based on a comparison principle for sub/super-solutions of the semilinear ODE (3.2) and a detailed understanding of the asymptotic behavior of the solution at 0 and∞(see [5]). ¹

1 Recently, X. Lamy [7] showed the uniqueness and monotonicity of the energy minimizing solution of equation (1.4) for the particular case ofF(u) given by (1.5) and on bounded domains by using a different technique.

4. Stability of the melting hedgehog

The goal of this section is to sketch the proof of Theorem 1.1. For that, we study the sign of the second variationQ at the melting hedgehogH defined at (1.6). The idea is to use a special basis decomposition forV ∈C_c^∞(R³;S0) and reduce the study ofQ(V) to simpler functionals by separation of variables.

4.1. Decomposition with respect to a special basis

We decomposeV ∈C_c^∞(R³;S0) in a certain orthogonal frame of S0. We use spherical coordinates:

x=r(sinθ cosϕ,sinθsinϕ,cosθ)∈R³, r∈R⁺, θ∈(0, π), ϕ∈[0,2π], and the following orthonormal basis inR³:

n= (sinθ cosϕ,sinθ sinϕ,cosθ), m= (cosθcosϕ,cosθ sinϕ,−sinθ), p= (sinϕ,−cosϕ,0).

Now we choose the following orthogonal frame inS0:

E0= ¯H, E1=n⊗p+p⊗n, E2=n⊗m+m⊗n, E3=m⊗p+p⊗m, E4=m⊗m−p⊗p.

Note that Ei =Ei(θ, ϕ) are independent ofr; moreover, {E1, E2} (resp., {E0, E3, E4}) form a basis of the tangent space (resp., the normal space) of the limit target manifold M at s+H¯ (defined at (2.1)).

Then anyS0-valued mapV can be represented as the following linear combination:

V =

4

X

i=0

wi(r, θ, φ)Ei. (4.1)

with{wi}i=0,...,4scalar functions onR³. 4.2. Instability fora² large

The ansatz consists in focusing on axially symmetric perturbations in (4.1), i.e., wi = wi(r, θ) are ϕ-independent, and moreover, studying the stability only on the subspaces {wiEi : wi = wi(r, θ) ∈ C_c^∞(R³)}. We obtain stability under such perturbations driven by the uniaxial basis componentsE0,E1

andE2, while the stability under perturbations driven by the biaxial basis componentsE3 andE4holds true for smalla² and fails fora² large (see [6]). We only present here the instability result:

Proposition 4.1 There existsa²₁ >0 such that if a² > a²₁ then one can findw∈C_c^∞(R³) independent of theϕ-variable with Q(wEi)<0 fori= 3,4.

We prove Proposition 4.1 in the special caseb²= 0 corresponding toa²=∞(see Remark 1). For that, we choosew(r, θ) =w_∗(r) sin²θ. Denotingf(u) = ^F(u)_u ^(1.5)= −a²+^2c2₃^u2, we compute:

Q(wE₃) =Q(wE₄) =32π 15

Z ∞ 0

h|w⁰_∗|²+ 4

r²+f(u)

|w∗|²i

r²dr=: 32π 15 E(w_∗).

Decomposingw_∗(r) =u(r)˚w(r) forusolving (1.4) and ˚w∈C_c^∞(0,∞), integration by parts yields:

E(w_∗) = Z ∞

0

n|uw˚⁰+u⁰w|˚²+ 4

r²u²w˚²+f(u)u²w˚²o r²dr

= Z ∞

0

n|uw˚⁰+u⁰w|˚²+ 4

r²u²w˚²+ (u⁰⁰+2 ru⁰− 6

r²u)uw˚²o r²dr=

Z ∞ 0

n|˚w⁰|²− 2 r²w˚²o

u²r²dr.

By (3.3), for small >0 there existsR >0 such that s+(1−)< u(r)< s+ on (R,∞). Since the best constant of Hardy’s inequality² inR³ is ¹₄ <2(1−)², one can find³ w˚∈C_c^∞(R,∞) withE(uw)˚ <0.

4.3. Stability fora² small

Let us briefly explain the strategy of proving stability of the melting hedgehog for smalla²>0 stated in Theorem 1.1. The first step is to reduce the study of the second variationQ to the axially symmetric context ofϕ-independent components{wi}_0≤i≤4 in (4.1). Indeed, as in the proof of the stability of the GL vortex (see [10]), we use Fourier decomposition inϕof the components{wi}_0≤i≤4; we show that the non-negativity ofQ(V) reduces to proving the non-negativity of three functionals Φk (k= 0,1,2) that correspond to the first three Fourier modes and each Φk depends only on three functions{vi(r, θ)}i=0,1,2

(i.e., vi are ϕ-independent functions). The second step is to reduce the study of the non-negativity of Φk to some new functionals defined onθ-independent componentsvi=vi(r). This separation of variable is much more subtle and uses some orthonormal bases of eigenvectors (depending only on θ) of certain operators coming from the Euler-Lagrange equations associated to the functionals Φ_k. Roughly speaking, this spectral decomposition is similar to sets of spherical harmonics, adapted differently for each of the v_i’s. The last step is to prove non-negativity of the remaining functionals depending only on components v_i =v_i(r). The strategy here relies on a Hardy decomposition (as in Section 4.2) that takes care of the nonlinearity of the problem coming from the bulk energy density.

Acknowledgements

The authors gratefully acknowledge the hospitality of Hausdorff Research Institute for Mathematics, Bonn where part of this work was carried out. R.I. acknowledges partial support by the ANR project ANR-10-JCJC 0106. V.S. acknowledges support by EPSRC grant EP/I028714/1.

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2 The standard Hardy inequality inR³:R

R³|∇ψ|²dx≥¹

4

R

R³

|ψ|²

|x|²dxfor everyψ∈C_c^∞(R³).

3 For instance, takingn large enough and definingψ :R+ →R+ by ψ(r) = _R¹ −¹_r on (R,^2nR_n+1), ψ(r) = ¹_r − _nR¹ on (^2nR_n+1, nR) andψ(r) = 0 elsewhere, we can choose ˚wto be a smooth (compactly supported) approximation ofψ.